Dyadic A1 weights and equimeasurable rearrangements of functions

TL;DR

This paper establishes that the decreasing rearrangement of dyadic A1 weights maintains an A1 property with a controlled constant, and proves the sharpness of this result within the class of such weights.

Contribution

It demonstrates that the rearrangement of dyadic A1 weights preserves the A1 condition with an explicit bound, and confirms the bound's optimality.

Findings

Rearranged dyadic A1 weights are A1 weights with a specific constant.

The A1 constant of rearranged weights is at most kc - k + 1.

The result's sharpness is proven for all such weights.

Abstract

We prove that the decreasing rearrangement of a dyadic A1 weight w with dyadic A1 constant [w]_{1,T}=c with respect to a tree T of homogeneity k,on a non-atomic probability space, is a usual A1 weight on (0,1] with A1 constant not more than kc-k+1.We prove also that the result is sharp, when one considers all such weights w.